© Copyright 2009 Dr. Ove Bratland
All rights reserved. No proportion of this book may be reproduced in any form or by any means, including electronic storage and retrivial systems, except by explicit, prior written permission from Dr. Ove Bratland except for brief passages excerpted for review and critical purposes.
“Intellectuals solve problems, geniuses prevent them.”
Albert Einstein
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Albert Einstein‟s wisdom regarding preventing problems before they occur certainly makes sense in pipeline and pipe network projects. Flow assurance – making sure the fluid flows as intended – relies heavily on mathematical models and the simulations they enable. Simulating the flow and everything affecting it contributes to problem prevention and efficiency, from feasibility studies through detailed engineering to operation. Ever more pipelines are being built around the world, and the number of people involved in various pipe flow calculations seems to increase daily. It is my hope that this book can be of help to everyone engaged in those tasks.
There are many commercial simulation tools available on the market, and the variation in user friendliness and underlying theoretical foundation for the various programs are astonishing. The purpose of this book is to explain how pipe flow simulation programs work and how to check results they produce. It goes into enough detail to enable the reader to create his own simulation tools and it also explains how to select and use commercial programs. It demonstrates some common sources of errors and how to avoid them.
Pipe flow is a complex phenomenon, and there have been a lot of new, valuable developments lately. Recent advancements come from such fields as fluid mechanics, mechanical engineering, chemistry, numerical mathematics, software development, control theory, and standardization. It is a challenge to keep up with it all, and this book intends to make the effort more manageable. The task is as much as possible seen from the engineer‟s point of view, and I have tried to avoid going too deep into details in the underlying theory.
Pipe flow problems can be categorized according to what sort of fluids we are dealing with, such as liquids, gases, dry bulk, or a mixture of several of them. This book is primarily about single-phase flow, meaning it focuses on pipes carrying either a liquid or a gas, but not both at the same time. It is still taking multi-phase flow into account in two important respects, though. It includes multi-phase simulation programs in the overview over different relevant commercial software tools in chapter 1, and it uses
transients. For readers who progress to multi-phase transient flow, the added equations required to do so will appear as a natural extension of the theory in this book.
In a typical pipeline project an oil company may be the project owner, while a contractor is used to carry out various phases of project execution. The contractor may do simulations in-house as part of this process, or he can sub-contract it to a company specializing in flow assurance. Results coming out of such simulations need to be verified as reliably as possible. Traditionally, this is done by using several subcontractors to do the same simulations and compare results. That can be very useful, but there are other, less well known ways of verification as well. A number of convenient verification tests have been presented in chapters 7.4.2 and 14.6, some published for the first time. The tests are meant to be useful to everyone involved in checking simulation results, including those who carry out the simulations in the first place. Given how easy some of the checks are, it does in fact seem natural to make such verification part of the contractual requirements.
A pipeline‟s capacity is one of the most important parameters in any design specification, and it is crucial to determine the friction accurately in order to meet that capacity as cheaply and reliably as possible. The most accepted way to determine the friction factor has been to use the traditional Moody diagram or the AGA calculation method. This book demonstrates that these traditional methods easily lead to 10 % inaccuracies in the pressure drop calculations, in some cases significantly more.
The traditional friction calculations suffer from two main weaknesses. First, they rely on measurements which do not stretch into as high Reynolds numbers as one may encounter (in high pressure export gas pipelines, for instance). Second, they rely on summarizing everything to do with surface texture into an „equivalent sand grain roughness‟. An overwhelming amount of measurements show this not to give accurate results in part of the relevant Reynolds number range.
Recently published measurements also show that coating can have significant effect on capacity, so much so that internally coated pipelines can achieve the same capacity with a significantly smaller diameter than similar uncoated pipelines. A large part of the book, all of chapter 2, is dedicated to showing how friction factor accuracies can be improved. Previously un-published diagrams are also given there. Some of the proposed methods rely on carrying out measurements and can be quite costly. When expensive pipelines are to be built, though, it makes sense to go into great detail regarding friction, and even early-phase laboratory measurements can be cost-effective.
The method of characteristics is probably the most used simulation method for liquid flow. It is fast, simple, and well known, but not directly applicable to gas flow. Chapter 7 outlines which simplifications the method of characteristics relies on, how to implement it in a computer program, and how to calculate steady-state starting values. Many steady-state methods have been developed over the years, but this book outlines a previously un-published method utilizing the transient simulation program modules to simplify the overall computer code.
Most books about transient gas pipe flow focus exclusively on how to simulate perfect gases. Real gases differ from perfect gases is some important respects, and perfect gas models are most useful as a reference for testing out simulation methods or for very low pressure pipes. Perfect gas models cannot be used in general simulation programs intended for both high and low pressure pipelines. Therefore, all gas theory in this book is developed with reference to real gases, and ideal gas models are used for reference or testing purposes only.
The fully transient gas model presented in chapter 10 uses the Kurganov-Tadmor scheme of order 3 in combination with an explicit fourth-order Runge-Kutta method to solve the conservation equations. The main focus is on how easy these methods are to use in practice rather than on presenting all the advanced theory they rely on. The KT2 method has been around for nearly ten years, but the high-order, causality-safe ways of dealing with boundary conditions and ghost cells outlined in chapters 12 and 13 has to my knowledge not been published before. The new methods make traditional simplifications redundant in some cases. Avoiding model simplifications increases the results‟ validity and applicability significantly.
Finally, some words about how both books are published. The traditional way of publishing goes via one of the established publishers, with all their resources for checking, editing, marketing, and sales. To most advisers‟ dismay, I have chosen not to follow that path. New technology makes it possible to handle most publishing tasks efficiently in alternative ways. Besides, the time when a book‟s content was married to the paper on which it was written is long gone, and the cost of making extra digital copies is zero. So why not let unpaid students get a digital copy for free. The same goes for those who want to consider the book for commercial purposes – just download the free version first and have a look. Orders for printed copies can be made at the internet site www.drbratland.com. Some of the simulation programs used in the examples can also be found there.
Any feedback from readers is greatly appreciated and should be directed to the internet site. All will be read, and as far as time allows, serious questions and comments will also be answered.
Ove Bratland February 2009
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The author wishes to thank the following companies for various discussions and support during the work with this book: Statoil, SINTEF Petroleum Research AS, AspenTech, Simsci-Esscor, Institute for Energy Technology (IFE), SPT Group, Institut Francais du Petrole (IFP), Telvent, Schlumberger, University of Tulsa, Neotechnology Consultants, Flowmaster and Advantica.
Thanks also to Prof. Gustavo Gioia for various discussions about the turbulence model in chapter 2.8, and to Dr. Elling Sletfjerding for discussions about his friction
measurements.
Professor Alexei Medovikov has given advice on how best to implement his DUMKA differential equation solvers, and warm thanks goes to him, too.
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Preface ... 3
1
Introduction ... 1
1.1 The many challenges involved in pipeline projects ... 1
1.1.1 History ... 1
1.1.2 Modern pipelines and their alternatives ... 2
1.1.3 Pipeline politics ... 2
1.1.4 What this book is about ... 3
1.2 Codes and specifications ... 4
1.3 A pipeline project’s different phases ... 4
1.3.1 Preliminary planning with feasibility study ... 5
1.3.2 Route selection ... 5
1.3.3 Acquisition of right-of-way ... 6
1.3.4 Various data collection... 6
1.3.5 Pipeline design ... 6
1.3.6 Legal permits and construction... 7
1.3.7 Commissioning and start-up ... 7
1.4 How pipe flow studies fit into a pipeline project, and which tools to use ... 7
1.5 Different sorts of pipe flow models and calculations ... 9
1.5.1 Single-phase versus multi-phase models ... 9
1.5.2 Steady-state versus transient simulations ... 10
1.5.3 The flow simulation software’s different parts ... 11
1.6 Considerations when simulating pipe flow ... 13
1.6.1 General considerations ... 13
1.6.2 Hydrates and wax ... 13
1.7 Commercially available simulation software ... 14
1.7.1 Single-phase pipe flow software ... 14
1.7.2 Steady-state multi-phase simulation programs ... 16
1.7.3 Transient simulation software ... 16
1.8 An example of what advanced pipe flow simulations can achieve ... 16
References ... 20
2
Pipe friction ... 21
2.1 Basic theory ... 21 2.1.1 Introduction ... 21 2.1.2 Laminar flow ... 22 2.1.3 Turbulent flow ... 242.2 Simple friction considerations ... 28
2.3 Nikuradse’s friction factor measurements ... 30
2.4 What surfaces look like ... 32
2.5 The traditional Moody diagram ... 36
2.6 Extracting more from Nikuradse’s measurements ... 40
2.7 The AGA friction factor formulation ... 46
2.8 Towards a better understanding of the friction in turbulent pipe flow ... 48
2.8.1 Introduction about turbulence ... 48
2.8.2 Quantifying turbulence ... 49
2.8.3 Using Kolmogorov’s theory to construct a Moody-like diagram ... 56
2.8.4 Comparing the theoretical results with other measurements ... 60
2.8.5 Large surface imperfections dominate on non-uniform surfaces ... 61
2.8.6 Friction behaves the same way for all Newtonian fluids. ... 63
2.9 Practical friction factor calculation methods ... 63
2.9.1 The surface-uniformity based modified Moody diagram ... 63
2.9.2 Improving friction factor calculation speed ... 67
2.10 Fitting curves to measurements ... 72
2.11 Friction factor accuracy ... 75
2.12 Tabulated surface roughness data ... 77
2.14 Transient friction ... 83
2.15 Other sorts of friction in straight, circular pipes ... 87
2.16 Friction factor summary ... 88
References ... 89
3
Friction in non-circular pipes ... 93
3.1 General ... 93
3.2 Partially-filled pipe ... 94
3.3 Rectangular pipe ... 97
3.4 Concentric annular cross-section ... 99
3.5 Elliptic cross-section ... 100
References ... 101
4
Friction losses in components ... 102
4.1 General ... 102
4.2 Valves ... 104
4.3 Bends ... 106
4.4 Welds joining pipe sections ... 108
4.5 Inlet loss ... 110
4.6 Diameter changes ... 111
4.7 Junctions ... 114
References ... 119
5
Non-Newtonian fluids and friction ... 121
5.1 Introduction ... 121
5.2 Pipe flow friction for power-law fluids ... 123
5.3 Pipe flow friction for Birmingham plastic fluids ... 127
5.4 Friction-reducing fluids ... 129 References ... 130
6
Transient flow ... 132
6.1 Mass conservation ... 132 6.2 Momentum conservation ... 135 6.3 Energy conservation... 1386.4.1 Sloping liquid pipeline with steady-state flow ... 142
6.4.2 Horizontal gas pipeline with isothermal steady-state flow ... 145
6.4.3 Example: Gas pipeline cooling down after stop ... 148
References ... 150
7
Simplified liquid flow solution ... 152
7.1 Main principles ... 152
7.1.1 General ... 152
7.1.2 Involving fluid properties ... 153
7.2 Solving the equations by the characteristics method ... 159
7.2.1 Example: Instantaneous valve closure ... 163
7.3 Boundary conditions in the method of characteristics ... 165
7.3.1 Pipe with constant pressure at the inlet, closed outlet ... 166
7.3.2 Pipe with valve at the outlet ... 166
7.3.3 Valve located any other place than inlet or outlet ... 168
7.3.4 Inline centrifugal pump ... 169
7.3.5 Pump between reservoir and pipe inlet ... 173
7.3.6 Positive displacement pump ... 173
7.3.7 Junction ... 174
7.4 Instantaneous valve closure ... 176
7.4.1 Basic simulations ... 176
7.4.2 Some ways to check the simulations results manually ... 179
7.5 Steady-state network analysis ... 180
7.5.1 General ... 180
7.5.2 Finding initial velocities using the steady-state characteristics method ... 182
7.5.3 Steady-state convergence criteria ... 184
7.5.4 Steady-state example... 185
7.6 Simulating transients in pipe networks, an example ... 188
7.7 Stability considerations ... 191
7.7.1 Frictionless flow ... 193
7.7.2 Flow with laminar friction ... 195
7.7.4 Some effects of the characteristic equations being nonlinear ... 200
7.8 Tracking the liquid ... 203
7.9 Checking simulation results ... 205
7.10 Advantages and limitations when using the method of characteristics ... 206
References ... 207
8
Heat exchange ... 209
8.1 General about heat through layered insulation ... 209
8.2 Heat transfer coefficient between fluid and pipe wall ... 212
8.3 Heat transfer coefficients for the pipe wall, coating and insulation layers ... 216
8.4 Heat transfer coefficient for outermost layer ... 217
8.4.1 Buried pipe ... 217
8.4.2 Above-ground pipe ... 218
8.5 The heat models’ limitations ... 221
8.5.1 Transient versus steady-state heat flow ... 221
8.5.2 Other accuracy considerations ... 222
References ... 222
9
Adding heat calculations to the characteristics method ... 224
9.1 The energy equation’s characteristic ... 224
9.2 Solving the energy equations using the explicit Lax-Wendroff’s method ... 229
9.3 Boundary conditions for the thermo equation ... 233
9.3.1 The problem with lack of neighboring grid-points at the boundary... 233
9.3.2 Junctions, pumps, valves and other components ... 235
9.4 Determining secondary variables ... 236
9.5 Computing starting values ... 237
9.6 Stability considerations for the energy solution ... 240
9.7 Numerical dissipation and dispersion ... 243
9.7.1 How numerical dissipation and dispersion can affect the simulations ... 243
9.7.2 Easy ways to reduce numerical dissipation and dispersion ... 245
9.7.3 Modern, effective ways to counter dissipation and dispersion... 247
References ... 254
10.1 Problem formulation ... 255
10.2 Some initial, simplified considerations ... 258
10.3 The conservation equations’ main properties ... 261
10.4 Selecting time integration and spatial discretization methods ... 265
10.5 How to account for friction and heat in the KT2 scheme ... 269
10.6 Calculating secondary from primary variables ... 273
10.7 Determining indirect fluid properties ... 276
References ... 278
11
Ghost cells ... 280
11.1 Some general considerations ... 280
11.2 Inserting ghost values: A simple method ... 281
11.3 An improved ghost cell approximation ... 284
11.4 Further ghost cell improvements ... 287
11.5 Computing state variables from flux variables ... 288
References ... 294
12
Boundary conditions ... 295
12.1 General ... 295
12.1.1 Boundary condition 1: Pressure source, inflowing fluid ... 296
12.1.2 Boundary condition 2: Pressure source, out-flowing fluid ... 297
12.1.3 Boundary condition 3: Mass flow source, in-flowing fluid ... 298
12.1.4 Boundary condition 4: Mass flow source, out-flowing fluid ... 299
12.2 Selecting boundary conditions in junctions ... 299
12.3 Other boundary conditions ... 301
References ... 302
13
Filling the ghost cells by using the boundary conditions directly ... 303
13.1 General philosophy ... 303
13.2 Mass flow source ... 305
13.2.1 Inflowing fluid ... 306
13.2.2 Outflowing fluid ... 307
13.3 Pressure source ... 308
14
Simulation results and program testing ... 310
14.1 Simulating one of the world’s longest gas pipelines... 310
14.2 Gas temperature in insulated pipelines ... 316
14.3 Simulating pipe rupture ... 318
14.4 How cooling affects the flow after shutdown... 320
14.5 Comparing with other simulation programs... 322
14.6 How to verify gas flow simulations, an overview ... 324
14.6.1 See if the integrations runs at all ... 324
14.6.2 Do the same checks as for liquid flow... 324
14.6.3 Checking the boundary and ghost cell approximations for steady-state flow ... 325
14.6.4 Checking the boundary and ghost cell approximations for transient flow... 326
14.6.5 Check that the program uses correct fluid properties ... 327
14.6.6 Check the heat flow calculations manually ... 328
14.6.7 Increase the velocity until choking occurs ... 328
14.6.8 Things which may confuse result interpretation ... 328
References ... 329
15
Simplified models ... 331
15.1 General ... 331
15.2 Steady-state calculations ... 332
15.3 Fully transient isothermal model ... 334
15.4 Neglecting part of the inertia for isothermal flow ... 335
15.5 Neglecting all terms to do with gas inertia ... 336
15.5.1 Model formulation ... 336
15.5.2 Numerical approximations ... 340
15.5.3 Important observations regarding neglecting the gas inertia ... 341
References ... 342
“Scientists discover the world that exists, engineers create the world that never was”.
Theodore von Karman
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This chapter presents some background information, including:
Pipeline history
How pipeline projects work
What flow simulations can be used for Different sorts of flow models
Single-phase versus multi-phase simulations
Overview of commercially available simulation programs
1.1 The many challenges involved in pipeline projects
1.1.1 History
Pipes appear to have been invented independently several places at nearly the same time and are known to have been in use as much as 5,000 years ago in China, Egypt, and the area presently known as Iraq. At a much later date, the Romans advanced the art of designing piping and waterworks, though the Roman empire‟s fall reversed all that and waterworks were largely ignored in early middle-age Europe. Towns reverted to using wells, springs, and rivers for water, and wastewater was simply disposed of into the streets. Improvements were clearly needed, and fittingly, one of the first books printed after the invention of the printing press in the fifteenth century was Frontinus'
Roman treatise on waterworks. The advent of the industrial revolution accelerated the
Pipes and channels have historically brought major advantages to those who had them, and successful pipeline or aqueduct projects have always required the right combination of political, economical and technical resources. History shows that most societies did not possess that combination, leaving them without advanced waterworks. Even today, a considerable part of the world‟s population suffers from unclean drinking water and inadequate sewage systems. The technology to solve such problems exists, but too often, poverty or economic unrest holds back the development.
1.1.2 Modern pipelines and their alternatives
In our modern world, pipelines have more applications than in previous times. They require relatively high initial investment and typically have a designed life-span of 40 years or more. That would probably not have impressed the ancient Romans, but it is still good enough to be more economical than alternative transport forms. Liquids can sometimes be cheaper to transport by ship, at least over long distances, but gas is difficult and expensive to transport in large quantities by any other means than pipelines. Gas can be liquefied, and Liquefied Natural Gas can be shipped long distances. To do so, however, a significant part of the gas‟ energy has to be spent on the liquefaction itself, and gas pipelines are generally the preferred option unless very long distances, difficult terrain, prohibitive legal regimes, or other special problems prevent them from being used.
1.1.3 Pipeline politics
Oil and gas pipelines can be very long, sometimes crossing country borders. Pipeline projects are often so important they get entangled in geopolitical complications, making long and careful negotiations with many interest groups an essential part of the project. Route selection is frequently dictated by environmental or political rather than technical concerns. High level politics was on daily display when this book‟s author stayed some years in Azerbaijan in the 1990s, during a time when a pipeline route from Azerbaijan via Georgia and Turkey to the Mediterranean Sea was selected in competition with
Even today, a considerable part of the world’s population suffers from unclean drinking water or inadequate sewage systems.
other, mostly cheaper alternatives. More than once, the amount of dignitaries visiting Baku in order to affect that and related decisions was so high that traffic flow in the city center suffered. For those managing the project at the time it must have felt like politics was everything and technology virtually nothing.
In some recent projects we have even seen that choosing relatively expensive subsea rather than overland routes have been motivated by desires to keep the number of parties involved at a minimum. Again, politics is more than a little involved.
At the time of this writing, an equally common and related problem faces the ASEAN countries (10 South-East Asian countries, including Indonesia, Malaysia and Thailand) in their efforts to expand their pipeline networks. Differences in national gas quality specifications make it hard to trade across borders: CO2-content can vary from nearly
30% to far less. This also complicates matters when securing backup alternatives in case of interruptions. One type of gas cannot always replace another even temporarily, and the added safety of having a gas pipeline network rather than one pipeline is reduced. Australia is an example of a country which has put intense effort into improving their gas quality standardization, and trade between different states goes more smoothly than it used to. Similar challenges related to gas quality standardization, customs for the gas as well as for spare parts, and a host of others are common all over the world wherever pipelines cross borders.
1.1.4 What this book is about
In addition to the geopolitical, environmental, and economical questions facing pipeline projects, there are myriads of interesting technical challenges to be solved as well. This book focuses on some of those technical challenges, specifically the ones to do with making the fluid flow the way it was intended. That is obviously affected by everything inside the pipe (inner diameter, surface roughness, and surface structure), fluid properties (there are lots of them, including viscosity, density, specific energy, and compressibility), and the pipe wall itself (thermo-properties, insulation, and elasticity). The environment affects the transported fluid‟s temperature, so submerged, buried and uncovered pipelines may have to be modeled slightly differently. The way the fluid flows is of course important to the pipelines‟ capacity, but also sets important conditions for phenomena that can damage the pipe: Corrosion, erosion, and the
Pipeline projects are often so important they get entangled in
potential formation of wax or other deposits fall into this category. Such damages fall outside the scope of this book, but the foundation for predicting them – the flow itself – is treated in detail.
It is easiest to deal with flow of the
single-phase type, meaning the fluid is either a homogeneous liquid or a gas, and that is what the book focuses on. But before going into details about pipe flow, let us have a brief look at some of the other aspects of relevance to pipeline projects.
1.2 Codes and specifications
A pipeline is always designed in accordance with codes and specifications. Those specifications describe nearly everything to do with the design, such as which materials to use, working stresses, seismic loads, thermal expansion, other imposed internal or external loads, as well as fabrication and installation. In addition, the design depends on factors relevant to the specific pipeline, including the fluid(s) to be transported (oil/gas/solids, single/multi-phase), length and required capacity, the environment (warm/cold climate, overland/buried/subsea, urban/countryside), and operational conditions (need for valves, compressors, pumps, surge chambers, storage capacity). Code compliance is mandated by various governmental organizations. Codes can be legal documents, and like other laws, they vary from place to place. Contractual agreements may typically also have a say on which codes to use, and all in all selecting the right codes and standards is often one of the most important parts of the project. The different relevant specifications typically overlap, and it is essential to decide what to do when that is the case, for instance that the most restrictive code applies. Many of the legal conflicts arising in large projects have to do with how different codes should be interpreted, or even more common, when to apply which code. Frankel (1996, 2002) gives an overview over different codes relevant to pipeline engineers, and more details can be found there. As a general rule, though, it is best to stick to international codes and standards as much as possible, and to minimize the use of company- or project-specifications.
1.3 A pipeline project’s different phases
The different phases in a pipeline project may vary considerably, depending on how large the project is, where it is, whether borders are crossed, whether the pipe goes over
This book focuses on technical challenges to do with making
the fluid flow the way it was intended.
land or subsea, who manages it and a multitude of other factors. The phases shown below can therefore only be seen as a typical example.
1.3.1 Preliminary planning with feasibility study
The main parameters are determined in this phase. They may include approximate pipe length with origin and destination, diameter, type of pipe, mass flow, capital cost, operating expenses with pressure loss and power consumption, main valves and pumping or compressor stations. Pipe flow simulations are very useful in this study. Both economical and technical feasibility should be considered. The project must be economical, and it obviously has to be technically possible. In addition, „political feasibility‟ is a major factor since conflicts and geopolitics can pose daunting challenges.
1.3.2 Route selection
For overland pipelines, the route should be marked on various sorts of maps. This can most often be done by using existing maps in addition to taking aerial photography and surveys of the pipeline route. Route maps and property plats are created from these. Right-of-way acquisitions are normally not done in this phase, but they are taken into consideration.
In case of rock tunnels, various additional sorts of surveys may be required, such as drilling to determine rock quality.
Existing maps are often of little help for subsea pipelines. Surveying can be quite complicated and expensive, but seafloor mapping technology has developed significantly in recent years. Maps and terrain models are generated using depth data from multi-beam echo sounders mounted on the hull of survey ships, and Remotely Operated Vehicles (ROVs) are also used. Autonomous Underwater Vehicles (AUVs) have been used in some recent projects and can be more economical and faster for some surveying tasks.
Many countries have strict laws prohibiting any activities from disturbing unexcavated archeological sites, and most project managers would surely prefer not to encounter any. But archeological sites can be stumbled upon almost anywhere. In a relatively recent development, The Ormen Lange-field off the Norwegian coast, a shipwreck was discovered, and archeological investigations had to be carried out before pipe lying. Needless to say, planning for such possibilities is not easy.
1.3.3 Acquisition of right-of-way
How this is done is to a large extent determined by local laws, and they differ a lot. The process can take the form of voluntary negotiation with land owners, or it can be condemnation, meaning the land is acquired through an involuntary legal process. Usually, owners are entitled to compensation at a fair market value. This can be a complicated, lengthy process with many involved parties. In this respect, subsea pipelines are the easiest ones to handle. As already explained, crossing borders generally complicates this task, sometimes to unmanageable levels.
1.3.4 Various data collection
This is similar to what was discussed under route selection, but the work is done in greater detail. Soil borings and various soil testing may in some cases only be possible after the acquisition of right-of way is finished, so it may have to be delayed until this phase.
1.3.5 Pipeline design
Because different industries use pipelines for different purposes, the design requirements are different and the types of pipe materials vary. In the petroleum and natural gas industry, steel pipe with welded joints is most common. Using high pressures steel pipes makes it possible to have fewer booster stations along the line, and steel‟s ductility enables it to bend and withstand considerable impact without fracturing.
In the water and sewer industries, on the other hand, pipes are normally under relatively low, sometimes atmospheric pressure. The low pressure has led these industries to prefer low-stress, non-corroding pipe materials as PVC and concrete. Both for low-pressure and subsea pipes, it is common for external loads to exceed the internal ones.
In the petroleum and natural gas industry, steel pipe with welded
1.3.6 Legal permits and construction
Once necessary legal permits and design are approved, construction can start. For overland pipelines, that may involve clearing a path of minimum 15 m, bringing in the pipe, possibly ditching, trenching, boring, tunneling, and river crossing, followed by welding, coating, wrapping, pipe laying, and backfill with restoration of land. For subsea pipelines, it means laying the pipe from the laying vessel, in some cases including building „underwater roads‟ or trenches, and to re-fill them after laying.
1.3.7 Commissioning and start-up
The various valves and instruments along the pipeline must be tested and found functional. There may be additional tests, too, such as pressure and leak tests, and various cleaning procedures may be necessary. For subsea pipelines, the fluid used to achieve the required buoyancy during lying must be removed. The procedures may include running cleaning and instrument pigs through the pipeline.
1.4 How pipe flow studies fit into a pipeline project, and
which tools to use
The whole purpose of constructing a pipeline is of course to have something flow through it, and understanding how the flow behaves is essential. Pipe flow simulation is used to optimize and verify design and to throw light on various operational issues. It is used not only through all the phases described in the previous chapters, but also for training engineers and operators. During pipeline operation, simulations are used for real time system estimation and forecasting, as well as for operator training. This book is about pipe flow, and it will show how the flow theory can help us to deal with all these tasks.
There are many pipe flow simulation tools commercially available (Bratland, 2008), but using them correctly and efficiently requires understanding of what the programs do, how they work, and their limitations. State of the art simulation tools are not good enough to be reliable if they are treated as „black boxes‟, and there is no substitute for understanding how they work in great detail. There is a danger that learning how to
simulate can be misunderstood as learning how to interface with simulation program A,
while it probably should mean something more like understanding simulation program
Figure 1.4.1. Various reasons to simulate pipe flow.
Considering all issues important to maintaining the fluid flow from inlet to outlet is sometimes called Flow Assurance. It is a term encountered frequently when studying pipe flow, particularly when hydrocarbons are involved. Still, there is no generally agreed on, clear, common definition of what Flow Assurance is. It is obviously possible to define the system boundaries inlet and outlet in different ways. For instance, when considering petroleum production, the inlet could be described as a reservoir or as one or several wells. Alternatively, it could simply mean the pipe inlet. The latter may have been the most common way to look at the problem in the past, but for gathering
Pipe flow
simulations
Feasability •Capacity •Single/multiphase •Insulation •Pumps, compress. •Oth. components Economy •Required componets •Power consump. •Capacity •Regularity Sizing •Pipe sections •Pumps •Compressors •Dampers Operation support •Training •Forcasting •'What if' •Planning Monitoring •Leak detection •Flow estimation •Hydrate & Wax •Spesial eventsnetworks, the trend for multi-phase simulation tools is towards integrated well and pipe network simulations. Following this trend, many of those involved in developing flow assurance tools are busy creating ever better interfaces so that almost any well simulator can communicate relatively seamlessly with any multi-phase pipe flow simulation package. The same can be said about the outlet end of the pipeline. The trend is to integrate with slug catchers, separators, processing facilities or whatever else the system contains.
The complexity of computing pipe flow depends on what the pipe transports and what sort of phenomena we want to investigate. Figure 1.4.2 illustrates some of the different parameters affecting how complicated it is to do those computations, arranged so that the simplest alternatives are on top.
Figure 1.4.2. Various parameters affecting pipe flow computation complexity
1.5 Different sorts of pipe flow models and calculations
The simplest way to classify pipe flow models is probably by specifying how many separate fluids they can deal with simultaneously (single-phase, two-phase or three-phase), and by whether they are able to describe time-dependent phenomena (transient or purely steady-state). Let us have a look at what these differences mean in practice.1.5.1 Single-phase versus multi-phase models
The first pipe flow models dealt with single-phase flow of water or steam, though not both at the same time. Since many phenomena are multi-phase, such single-phase models have their limitations. Early studies on transient two-phase flow were conducted in the nuclear industry, as it became mandatory to predict the transient flow behavior during potential Loss-of-Coolant Accidents for licensing pressurized water reactors.
Phases
Singlephase MultiphaseTime
Steady-state TransientFluid
Single- component Multi-componentThermo
Isothermal Heat lawsSystem
Single pipe NetworkSpeed
Offline Real-timeInterface
Non-standard StandardMulti-phase flow can also occur in gas pipelines. If even a small amount of liquid condenses on the pipe wall, it will affect the flow. As we will see in later chapters, a gas pipeline‟s capacity can be very sensitive to the wall surface roughness, and it takes only a tiny amount of droplets on the wall to affect the friction significantly. It is essential to know whether condensate forms or not, and dew point specification is frequently part of gas sales contracts. If a small amount of condensate is present, one may get away with simply modifying the friction factor while keeping a single phase model and still get reasonably accurate simulation results. If the amount of condensate gets larger, computations based on single-phase models can no longer do the job. In some cases it is clear from the start that the flow can only be modeled sensibly with multi-phase software. That is the situation when we want to simulate a well flow of oil, gas and water mixed together. Slugging, a common problem, is very much a multi-phase phenomenon, and flow models may be used to investigate how high the gas velocity needs to be to avoid it. Predicting such operational limits, the flow envelope, calls for multi-phase simulations.
1.5.2 Steady-state versus transient simulations
Some commercially available software packages are steady-state, meaning they can only tell how the pressure, flow, and in some cases temperature, is going to be distributed along the pipe(s) once some sort of equilibrium state has been established. They cannot tell us how conditions are on the way to that equilibrium. We see that already in the definition of a steady-state simulator some of its limitations become apparent: It cannot describe transient phenomena like line packing or pressure surges, nor can it produce a meaningful result if the system itself is unstable and therefore never converges towards a steady state. A fully transient simulator, on the other hand, computes all intermediary steps on the way to the new steady-state when such a state exists. That means transient simulations produce more information, but at the cost of using more CPU-time.
Transient programs need some steady-state solver integrated, either in the form of separate steady-state program or by mathematically solving the transient equations for
A steady-state simulation program cannot describe transient phenomena like line packing or pressure surges. Nor can it produce a meaningful
result if the system itself is unstable and therefore never converges towards a steady state.
the time derivative being zero. Many of the transient phenomena of interest are simulated using a steady-state situation as a starting point, so transient simulations may rely on steady-state computations in order to define the initial condition on which the transient simulations should be based.
1.5.3 The flow simulation software’s different parts
Figure 1.5.1 illustrates some of the main parts a simulation program may include. A commercial program package have several separate parts, it may require several licenses and may also rely on many software and hardware interfaces. Even the simplest possible simulation program must at least provide a way to give input data, typically via a Graphical User Interface (GUI). It must know the chemical/physical properties of the fluid(s) involved (PVT-data), and it must contain a computation module. It needs a way to communicate results, for instance via the GUI or via an
Application Programming Interface (API) with another program.
Figure 1.5.1. Typical flow simulation software structure (simplified).
Simulating a straight pipe containing water can be done with a program containing less than 10 lines of code. Adding all whistles and bells necessary to make the program flexible and user friendly, those 10 lines grow to many thousands. When well structured, the program parts do not all have to come from the same developer. Therefore, the different modules need convenient, preferably standard ways to talk to each other, and also to talk to the outside world. Lots of effort goes into making
Main calculations API
Steady-state regime Flow
Thermo Transient
GUI
different program modules integrate well on Internet Protocol (TCP/IP), various
Microsoft’s technologies (DCOM and later .NET) and industry standards (CAPE-OPEN, 2003, and OPC.)
Note that the way programs are structured and which main modules they contain are the same whether the program computes single- or multi-phase flow, steady-state or transient. For instance, Simsci-Esscor‟s PipePhase contains one module for multi-phase steady-state simulations, and it integrates with TACITE for multi-phase transient simulations. The user interface is not much affected by the TACITE integration (but the price is!). Similarly, the same computation modules, say OLGA, can be used with many different simulation packages, even though the license typically has to be bought separately.
Computation modules vary between different programs. They generally contain fluid flow equation solvers, and they may contain one or several thermal models. For multi-phase flow, there is also some sort of flow regime identification software. That determines whether the flow is annular, bubbly, slug, or of another type. Today‟s multi-phase software varies somewhat in the way they determine the flow regime in each part of the pipe, but they all rely heavily on empirical data. At the same time, all multi-phase simulators are very sensitive to getting the flow regime right, even though that is one of the least accurate part of the programs.
The thermal models in use vary greatly, from the simplest isothermal models to detailed transient models of the heat flow both in the fluid, pipe wall and surroundings. The thermal model in chapter 8 discusses this in greater detail.
There is also much variation in how different programs handle PVT-data. In a water pipeline, one may get reasonable results by simply specifying the water‟s density, compressibility and viscosity as three constants. Those properties are in reality not constant but vary with temperature and pressure, and an improved model needs to know how those properties are related. It also makes sense to include vapor pressure data to enable the program to give warning in case of cavitation. In systems where cavitation is permitted, the program may be expected to compute exactly how the resulting 2-phase water/steam mixture behaves, and hence PVT-data needs to be available for steam as well. In addition, specific heat and surface tension must be known in order to include heat and flow regime estimation. Some fluids are much more complex than water, and several vendors have specialized in developing PVT-data packages. At the time of this writing, the most used commercially available such
packages seem to be the AGA Program, Gaspack™, GasVLe, Aspen HYSYS®, Multiflash, PRO/II, PVTp and PVTSim (Bratland, 2008).
Note that a simulation program must update PVT-data in all grid-points as the pressure and temperature change during computation. This means the computation module has to talk to the PVT-module continuously, and experience show that the PVT-data module easily ends up taking most of the computer‟s capacity. The simulation program may alternatively read out necessary data first and tabulate them for fast lock-up later, but that introduces its own problems. Since one of the main challenges when creating pipe flow simulation modules is to make the program fast enough, it is important for the PVT-data to be handled efficiently.
1.6 Considerations when simulating pipe flow
1.6.1 General considerations
Early phase concept studies may permit relatively inaccurate computations, in some cases favoring steady-state software over
more detailed transient simulations. Note, though, that using the same software through as many phases as possible reduces the need to familiarize with many different interfaces, and depending on how the model is built up, it can also save work. The
model should generally be built in several steps, starting by simulating a simplified system. It is best to neglect all nonessential components during the first runs, and get a feel for how the system is performing. Using automated routines for feeding all component data from CAD-drawings into the simulation model, as some software vendors seem to suggest, rarely makes sense, particularly not in an early phase. Components should rather be added gradually while running increasingly sophisticated simulations. Deciding which details to include and where to simplify is an important part of model building, and it happens to be a kind of task humans tend to be better at than computers.
1.6.2 Hydrates and wax
Hydrates are ice-like structures which form when water and natural gas are in contact at high pressure and low temperature. Paraffins in crude oil or condensate can lead to
The PVT-data module easily ends up taking most of the
phenomena depend on pressure, temperature, chemical properties, and fluid velocity. Although recent progress has been made in cold hydrate pipe flow technology, avoiding hydrates and wax for the most part comes down to keeping the flow relatively hot and/or injecting inhibitors like methanol or glycol. Multi-phase simulations may be used to study how to avoid problems with hydrates and wax, and to some extent how to deal with them if they occur. Since avoiding problems with depositions can be expensive, it pays to use as good flow and thermal models as possible for such studies.
1.6.3 Leak detection
Using simulation-based leak detection systems is also becoming increasingly popular and some companies‟ market software modules for that specific purpose. Two different detection principles are currently in use: Neural network-based decision making and calculations based on flow models. Implementing a leak detection system involves studies of how accurately various sorts of leaks can be detected by the chosen method when fed by signals from available sensors. The required leak detection accuracy has an impact on the system‟s complexity and costs. Deciding which accuracy to target is a significant part of deciding what to install. Note also that the implementation phase has not always been completely successful in previous leak detection projects. It is crucial to bring all the concerned parties on board early in system planning, design and testing, and also while developing appropriate operational procedures.
1.6.4 Other features
Simulation tools may also be used for operator training and various system testing. Such software is used for operations as varied as pigging, erosion control, corrosion control, sand buildup studies, and nearly any other phenomena related to fluid flow. Again, deciding to which extent those are central issues is something to consider before deciding which details the software needs to take into account in order to satisfy ones requirements.
1.7 Commercially available simulation software
1.7.1 Single-phase pipe flow software
A simple internet search using terms like flow assurance or pipeline simulation software produces hundreds of thousands of hits. Not all of the hits are unique, and not all have to do with pipeline simulation programs, but it is still easy to see that there are lots of
alternatives available. The vast majority of those programs can only simulate single-phase flow. Prices range from 0 (free!) to thousands of dollars. Given that enormous diversity no attempt has been made to give an extensive overview of the different
Name Contact Comments
Stoner Pipeline Simulator
Advantica
www.advanticastoner.com
Large simulation package with many modules and support offices around the world. Relies on built-in PVT-data. Flowmaster Flowmaster Ltd
flowmaster.com
Integrates with Matlab. Both liquid and gas. Also thermo modules. Does not focus on systems where relatively complex PVT-data are required.
Atmos Pipeline Software
Atmos atmosi.com
Involved in all sorts of singe-phase pipeline computations. Offices or representatives in 28 countries.
GASWorkS Bradley B. Bean b3pe.com
One of the many cheap of-the-shelf steady-state gas networks simulators. Developed by a competent, but very small company.
FluidFlow3 Flite Software fluidflowinfo.com
Both gas and liquid simulations. Comes with 850 pre-defined fluids in its database. Can also handle Non-Newtonian fluids.
AFT Pipeline Applied Flow Technology aft.com
Well designed, modularized steady-state and transient software. Has separate module for PVT-data.
PipelineStudio Energy Solutions www.energy-solutions.com
Extensive collection of software modules for design, analysis, optimizing and forecasting oil and gas networks.
FlowDesk Gregg Engineering greggengineering.com
Gas pipeline simulator. Focuses a lot on scheduling and forecasting.
SIMONE Liwacom
liwacom.de
Simulation and optimization of natural gas pipeline systems.
H2OCalc MWH Soft
mwhsoft.com
Specialize in various types of water pipeline computations.
Table 1.7.1. Single-phase pipe flow simulation software
software in this category, and table 1.7.1 should in no way be considered complete. Instead, it intends to illustrate that different software serves different market niches, even though they are mainly built on the same well-known theory. The most important
requirements properly, contacting a vendor, and discussing how those requirements can be met.
1.7.2 Steady-state multi-phase simulation programs
The steady-state programs are generally relatively easy to use, and they are probably used more than the transient programs. Nearly all multi-phase simulators focus on some sort of transient capabilities, such as their ability to integrate with a third-party transient simulator. That is a strong indication that the developers recognize a trend towards transient simulations.
1.7.3 Transient simulation software
OLGA is today probably the most well documented and advanced multi-phase transient pipe flow simulator on the market, but there are also others, see table 1.7.3. Additional multi-phase transient software packages are under development, and some of the existing ones are being improved. Interestingly, some of the oil companies
sponsor several of the development projects at the same time (Bratland, 2008).
1.8 An example of what advanced pipe flow simulations
can achieve
Ormen Lange is at the time of this writing (2008) the largest natural gas field under
development in the Norwegian continental shelf. The field is situated 120 km northwest of Kristiansund, where seabed depths vary between 800 and 1,100 meters. The reservoir is approximately 40 km long and 8 km wide, and lies about 3,000 meters below sea level. The Gas production is planned to become 60∙106 m3/day once full capacity is reached.
Using offshore separation of gas and liquids produced from the reservoir would have been a relatively conventional, but also expensive way to develop the project. It was concluded that offshore separation could be avoided and that the produced multi-phase flow could be sent to shore through pipelines directly. For this to be feasible, an
Using multiphase flow to send produced gas, oil and water to shore directly can be much cheaper than offshore separation.
advanced flow assurance solution was required.
Name Contact Comments
HYSYS Pipe Segment AspenTech aspentech.com
Not a very extensive model. AspenTech recommends other software for more advanced export pipelines, gathering systems or riser analysis.
HYSYS PIPESYS AspenTech aspentech.com
Licensed separately from the Hysys Process simulation package. More advanced than Hysys Pipe Segment and used for pipeline design and analysis. PIPESIM Schlumberger
www.slb.com
One of the most well known and most used simulation packages for multi-phase pipe flow. Developed to integrate nicely with the well simulator Eclispe. Both 2- and 3-phase.
GAP Petroleum Experts
petex.com
Part of the Integrated Production Modelling Package, which also includes various well simulation software. Both 2 and 3-phase.
PROFES Aspen Tech
aspentech.com
Dynamic multi-phase models that can be implemented within the Aspen HYSYS environment. Both 2 and 3-phase. When the Profes Transient module is included, it can also perform transient analysis.
PIPEPHASE Simsci-Esscor (Now owned by Invensys) www.simsci-esscor.com
Developed for simulation of complex networks of pipelines and wells. Both 2 and 3-phase. Can be licensed with the TACITE transient module as an integrated part.
PIPEFLO Neotechnology Consultants Ltd. neotec.com
One of the veteran steady-state multi-phase simulators. Comes with 2-multi-phase capabilities.
TUFFP Pro University of Tulsa www.tuffp.utulsa.edu
This software is integrated into PIPEPHASE and PIPESIM, but also used separately. Both 2- and 3-phase.
DPDL University of Tulsa www.tuffp.utulsa.edu
Two-phase liquid-gas isothermal flow. Very cheap, comes with Shoham’s book (Shoham, 2006). Well documented in the book.
An integrated flow assurance system based on the OLGA multi-phase simulator has now been installed and is in daily use. As described by Aarvik et al., (2007), it includes five sub-systems: The Pipeline Management System, the Virtual Flow Meter System, the Production Choke Control System, the Monoethyleneglycol (MEG) Injection Monitoring and Control System, and the Formation Water Monitoring System. The underlying models start at the reservoir influx zone, and include detailed representations for the subsea wells and templates, production pipelines and on-shore slug catchers. The operator is given access to liquid monitoring data throughout the system and receives recommendations on such vital parameters as choke set points and MEG injection rates. Another important feature is that the system serves as redundancy for the multi-phase flow meters. If and when the wet gas meters fail, useful flow data for each well is still going to be available from the estimates produced by the Virtual Flow Meter System. The flow assurance system can run in four different execution modes: Real Time System Mode, Look-ahead Execution Mode, Trial Execution Mode, and Planning Execution Mode. This flexibility gives operators and planners a wide range of ways to improve their procedures and investigate „what if‟-scenarios.
Figure 1.8.1. Overview of Ormen Lange subsea production system. © Norsk Hydro.
Name Contact Comments
OLGA SPT Group
www.sptgroup.com
Currently the most used and also probably most well documented transient pipe flow simulation software. Handles both 2 and 3 phase flow. Integrates with the most used well and process simulators, in addition to most of the steady-state multi-phase pipe flow simulators.
TACITE Simsci-Esscor
www.simsci-esscor.com
Developed by Institut Francais du Petrole (IFP), but marketed by Simsci-Esscor as part of its PIPEPHASE package. Does not seem to have an open, documented API, and so can only be used together with PIPEPHASES’s Graphical User Interface. The current version does not have full network capabilities. Both 2 and 3 phase. SimSuite Pipeline Telvent
telvent.com
2-phase simulator originating in the nuclear industry, but used for both water/steam and oil/gas the last 10 years or so. It comes integrated with a steady-state simulator.
ProFES Transient Aspen Tech aspentech.com
Developed by AEA Technology in the UK, it was formerly known as PLAC, (based on TRAC, developed for the nuclear industry), later integrated into AspenTech’s ProFES simulation package to bring transient capabilities to its steady-state module. Development has been discontinued; the software is no longer marketed.
Aspen Traflow Aspen Tech aspentech.com
Originally developed for Shell but also used in other projects. No longer developed or marketed.
Table 1.7.3. Multi-phase transient pipe flow simulation software
After the gas has been processed onshore in Norway, it is exported to Britain through a
extensively in every stage of that pipeline project, too, both for selecting main pipeline parameters well as for all the other purposes mentioned in figure 1.4.1.
References
Frankel, M. (1996, 2002): Facility Piping Systems Handbook. Second Edition, McGraw-Hill. CO-LaN Consortium (2003): Documents 1.0 Documentation Set (freely available from colan.org).
OPC Foundation: Standards for open connectivity in industrial automation. (available from opcfoundation.org).
Ellul, I.R., Saether, G., Shippen, M.E. Goodreau, M.J. (2004): The Modelling of Multi-phase Systems under Steady-State and Transient Conditions – A Tutorial. Pipeline Simulation
Interest Group PSIG 0403.
Liu, H. (2005): Pipeline Engineering. Lewis Publishers.
Shoham, O. (2006): Mechanistic Modeling of Gas-Liquid Two-Phase Flow in Pipes. Society of Petroleum Engineers.
Bryn, P., Jasinski, J.W, Soreide, F. (2007): Ormen Lange – Pipelines and Shipwrecks.
Universitetsforlaget.
Aarvik, A., Olsen, I., Vannes, K., Havre, K., Kroght, E., C. (2007): Design and
development of the Ormen Lange flow assurance simulator, 13th International Conference on Multi-phase Production technology. p.47-64.
Bratland, O. (2008): Update on commercially available flow assurance software tools: What
they can and cannot do and how reliable they are. 4th Asian Pipeline Conference &
“Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the eddying currents, the other to the random and reverse motion.”
Leonardo da Vinci on turbulence 1490 AD
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This chapter outlines how to calculate friction in straight pipes:
Various ways to define the friction factor
Nikuradse‟s and Moody‟s traditional friction factor diagrams How surfaces affect friction
Surface roughness values for some typical surfaces
Recent improvements based on measurements and turbulence theory Friction factor accuracies
Putting it all together
2.1 Basic theory
2.1.1 Introduction
When fluid flows through a pipe, friction between the pipe wall and the fluid tries to slow down the fluid. Unless we get assistance from gravity or naturally occurring pressure, we generally have to install pumps or compressors to counter the friction. As one would expect, many researchers have investigated it and come up with practical ways to describe it. It turns out that even for single-phase flow, pipe friction is a complex phenomenon and questionable friction calculations are surprisingly common. In addition to nature-given difficulties, there are also some historical reasons for the
current confusion: The theory has evolved gradually over the years, though some outdated definitions and methods have survived and remain in use today. Even though pipe friction is very similar for gas pipelines, oil pipelines, blood vessels and even open channels, different calculation methods are currently in use for different types of pipes or fluids. That practice tends to complicate matters and is strongly discouraged in this book.
Loosely stated, pipe flow can be either laminar or turbulent, and the physics involved changes significantly when we go from one to the other. Closer inspection reveals that no such thing as completely turbulent pipe flow exists, there is always a laminar sub-layer closest to the wall. A pipe‟s surface properties become more important the more turbulent the flow gets. The traditional way of taking this into account has been by compressing the whole surface description into something called an equivalent sand
grain roughness. This approach has the advantage of being very simple, but we will soon
see that it can lead to rather inaccurate results.
Another important thing to remember is that most of the well-established methods for calculating pipe friction were only ever intended for steady-state flow. In transient flow, our steady-state friction theory is, strictly speaking, invalid. We therefore need to establish an understanding for which conditions we can expect the results to be acceptable under.
Since friction is a very important parameter in determining a pipeline‟s capacity, we are going to dedicate much effort to this subject, discussing the most common calculation methods and proposing some best practices. We are also going to show which accuracies we can expect for different sorts of calculations.
For those less concerned with exactly how the theory is developed, it may not be necessary to study all of chapter 2 in-depth. Instead, the resulting diagrams in figures 2.9.1-2.9.3, as well as chapters 2.11-2.16 should be of most interest.
2.1.2 Laminar flow
For steady-state single-phase flow, the Reynolds number Re can be used to determine whether the flow is fully laminar. Alternative definitions of Re are given in equations
Pipe friction is quite a complex phenomenon and questionable
friction calculations are surprisingly common.
2.1.8, 2.1.10, 2.2.5 and 3.1.1. At very low Re, typically below 2,300, it has been found that the flow tends to be fully laminar. For so-called Newtonian fluids, which include water, air, and most of the other fluids engineers have to deal with, Newton‟s law of viscosity (sometimes referred to as Stoke‟s law for laminar flow) states:
(2.1.1)
where τ [N/m2] is the sheer stress and v [m/s] is the velocity at distance y [m] from the
wall. µ [kg/(ms)] is the fluid‟s dynamic viscosity.
In a circular pipe, y is the distance from the pipe wall, and therefore y = d/2 – r, where d is pipe‟s inner diameter and r is radius from pipe center to the studied point.
Figure 2.1.1. Velocity profile. The velocity is zero at the wall and increases towards the center.
Figure 2.1.2. Shear forces on cylindrical fluid element.
During steady-state conditions, the friction shear force on any cylindrical fluid section of length l and radius r has to be balanced by an equal force due to the pressure difference Δp = p1 – p2 working on that cylinder‟s end sections:
